Inward motion of diamond nanoparticles inside an iron crystal

In the absence of externally applied mechanical loading, it would seem counterintuitive that a solid particle sitting on the surface of another solid could not only sink into the latter, but also continue its rigid-body motion towards the interior, reaching a depth as distant as thousands of times the particle diameter. Here, we demonstrate such a case using in situ microscopic as well as bulk experiments, in which diamond nanoparticles ~100 nm in size move into iron up to millimeter depth, at a temperature about half of the melting point of iron. Each diamond nanoparticle is nudged as a whole, in a displacive motion towards the iron interior, due to a local stress induced by the accumulation of iron atoms diffusing around the particle via a short and easy interfacial channel. Our discovery underscores an unusual mass transport mode in solids, in addition to the familiar diffusion of individual atoms.

Reviewer #1 (Remarks to the Author): Comment 1 The observed rigid body motion of diamond particles into bulk iron is indeed very interesting.However the presented theory is not convincing to me and lacks experimental evidence.A carbon concentration gradient and resulting chemical potential gradient is given as the origin of the driving force.This could be supported by measurements of the carbon concentration gradient (for example using SIMS).

Reply:
We are glad that the reviewer considers our finding is very interesting.We also highly appreciate the valuable suggestions given by the reviewer.A carbon concentration gradient from the surface towards the interior of bulk Fe has been evidenced by the depth profiles and three-dimensional-compositional images of C and Fe from the timeof-flight secondary-ion mass spectroscopy analysis (ToF-SIMS, Fig. 4b).

Fig. 4b
ToF-SIMS analysis of a sample heated in furnace for 1 hour (all dispersed DNPs have entered the iron matrix) and then quenched in water to retain the carbon distribution at high temperatures.The ToF-SIMS spectra showing the depth profiles of secondary ions of C -(grey) and Fe -(blue) in the sputtered volume from the top surface to a ~60 m depth over an area of ~5050 m 2 .The corresponding 3D images of the depth profiles visualizing the opposite concentration gradients of C and Fe.The inset ToF-SIMS spectrum displays the depth profile of C -in a much smaller volume of the sample (the analyzed area is ~55 m 2 and the depth is ~ 500 nm).

Action taken
We have added the SIMS characterization in Fig. 4b.

Comment 2
The chemical potential gradient applied in the modeling is extreme, by many orders of magnitude.The authors claim that the particle trajectories can have millimeter length.
Along these distances the concentration (and chemical potential) gradient would be very shallow.I'm also skeptical that the solution of carbon from the DNPs would allow for millimeter travel.Maybe a rough estimate of the resulting migration speed would be helpful here.

Reply:
Diamond nanoparticles at the millimeter-depth of the Fe matrix have indeed been found (verified by the Raman spectroscopy, Fig. 3c), though the signal intensity of diamond decreases significantly as the depth increases (only some leading DNPs with high enough local concentration gradient can reach the millimeter depth).In the following, we make an explanation about how the local concentration gradient is built up and sustained.
Here the source of C is the graphitized surface layer of each and every DNP gradually dissolving to satisfy the solubility in the surrounding Fe.The carbon concentration profile could be maintained for a long time via the continual supply from these DNPs (the number of DNPs entering Fe is sufficiently large).Note that different from the conventional carburizing, in which the carbon source only exists outside the steel and the carbon concentration gradient becomes shallower and shallower along the millimeter depth direction, here each engulfed DNP with a gradually graphitized and meanwhile dissolving interfacial layer acts as a movable source of carbon by itself.As a result, the large carbon concentration gradient extends forward along with the moving DNPs (inhomogeneous concentration field).The driving force for the inward motion of each DNP is the local chemical potential gradient across the nanoparticle itself instead of across the long millimeter-distance (we have redrawn the schematic diagram in Fig. 4a to highlight this).
Take the leading DNP as an example, as it moves forward deeper and deeper, it keeps encountering iron interior with less carbon solute content (i.e., the DNP always sees a higher Fe concentration at the location in front of its moving trajectory).The dissolved carbon atoms at the location underneath the DNP diffuse into the "virgin" iron more rapidly than into the already-carbon-enriched locations above the DNP.That is, in its surroundings the DNP is always accompanied by a local concentration gradient, see the C profile in a local (smaller volume) region, likely around some DNPs, as displayed in the inset of the TOF-SIMS spectra, Fig. 4b.
As a result, the carbon concentration gradient extends forward along with the moving DNPs (inhomogeneous concentration field), and makes the millimeter travel of DNPs possible.
We also have performed a back-of-the-envelope numerical estimate of the travel velocity of the DNP.The DNP is treated as a sphere with diameter d, and the Fe volume flux JcV (in the unit of m 3 /s) through the equatorial plane of the diamond sphere can be written as (2) where dc/dz is the Fe concentration gradient (or the negative carbon concentration gradient, in units of atoms/m 3 /m) across the DNP, is the effective diffusivity of Fe atoms at the Fe-DNP interface, s is the thickness of Fe atoms flux and equals the diameter of a Fe atom, about 0.25 nm.The translational movement of the DNP fills in the space left by leaving Fe atoms, and the volume flux of iron atoms JV can also be written as where the translation velocity  can be expressed as: where =610 -11 m 2 /s (see the Supplementary discussion II and III for details).For a DNP with d = 100 nm, with the maximum chemical concentration difference across its two poles (at 1,250 K, the maximum carbon concentration reaches 1.8 wt.%, i.e., close to 7 at.%, see Fig. S8), Equation (4) predicts a maximum v ~120 nm/s.This estimated DNP velocity is comparable with that observed in experiments (the DNPs reach up to ~1 mm within 5 h, the maximum motion velocity is hence ~0.2 mm/h or ~50 nm/s).

Action taken
We have added the discussion about the local concentration gradient in the main text (Page 6), and highlighted it in Fig. 4a.We also have supplemented the back-of-theenvelope numerical estimate of the velocity of DNP in Supporting Information.

Comment 3
The supplemental videos look great, but are not conclusive either (is the particle really entering the foil, or migrating along the surface?).I wonder if the motion of the particle leaves behind a disordered channel that could promote fast diffusion of iron atoms toward the surface.Either way, something has to break the symmetry and help the particle sustain the inward direction.Further experimental evidence is needed in my opinion before this interesting result can be published.

Reply:
Diamond nanoparticles (DNPs) move into the Fe foil, rather than migrate along the surface.Please see the real-time observations of the sink-in process of DNPs recorded in an added video (Supplementary Video 2), where the yet non-entered particles on the surface of Fe serve as the reference in a fixed coordinate system.The right panel of this video demonstrates the real-time sink-in process of diamond nanoparticles under the secondary-electron imaging, which is simultaneously recorded with the scanning-TEM observations (the left panel).
We agree with the referee that the motion of the particle may leave behind a "disordered channel" that could promote fast diffusion of iron atoms toward the surface.To verify this, we have conducted an in-situ TEM experiment to trace the motion of the DNP cluster (marked in orange, Fig. R1a).After the DNP entered the Fe matrix, we stopped heating and then further thinned the sample to expose the engulfed DNP, as shown in the TEM image (Fig. R1b), from which we can see the contrast of defects (distortionof the crystal lattice) in the upper zone of the DNP indicating a possible easy channel left behind by the downward DNP for the Fe atoms diffusion at high temperatures.

Action taken
We have added this discussion on page 7: "Note that the downward motion of the DNP may leave behind an "easy channel" that could further promote the fast upward diffusion of the Fe flux towards the sample surface.Fe atoms leaving from the upper interface of Fe/DNP help to sustain the local concentration gradient around the DNP and hence continuous motion of the particle." Reviewer #2 (Remarks to the Author): Comment 1 The authors report an abnormal and remarkable mass transfer way in solids, i.e., robust diamond nanoparticles entering into solid iron/steel crystals as whole rather than decomposed individual atoms in the absence of externally applied mechanical force.The preservation of crystalline diamond structure inside iron is beyond our conventional wisdom that diamond is easily graphitized when comes into contact with iron, especially at elevated temperatures the inevitable chemical wear of diamond tools in cutting of steels.The underlying mechanism (involves dissolution of oxide scales on iron, surface transformation from diamond into graphite, and the chemical-potential gradients induced local stress) has been unraveled, using in-situ microscopy and bulk experiments as well as the modelling techniques.The thermodynamic and kinetic processes of the sinking-in and translation of diamond particles in iron have been taken into consideration and analysed by the authors in a very thorough manner, leading to a consistent explanation of their observations.Besides the scientific significance, this work provides a new method for nanodiamonds-dispersion strengthened steels with ultrahigh hardness and wear resistance, superior to what can be currently offered by the familiar powder metallurgy.

Reply:
We thank the reviewer for pointing out the significance of our work.

Comment 2
To give this work more implications for the development of diamond-based materials and devices, I would like to suggest the authors give a little bit more discussion to clarify the difference between this work and previous studies on diamond etching using other metals such as Ni.
Reply:  Communications in its present form.

Reply:
We really appreciate the referee's valuable suggestions and the recommended literature, which are very helpful for us to solidify the underlying mechanism.We have added experiments, analytical calculations, and revised the manuscript according to the referee's suggestions and cited the relevant work mentioned by the referee.In the following, we provide a point-to-point response to address the review questions.

Comment 2
The initial sinking of DNPs into subsurface region of iron is related to the good wetting of graphitized diamond by Fe.Similar observations have been reported in the past [1].
This initial sinking has nothing to do with the "Gibbs adsorption isotherm", as claimed in page 5 of the manuscript.

Reply:
We agree with the referee that the Fe flux spreads over the particles via fast surface diffusion, resulting in a high degree of wetting.The capillary force arises from the interaction between the Fe flux and the DNP surface, directed along the nanoparticle/Fe We have removed the discussion about "Gibbs adsorption isotherm", cited the work of

Zimmermann et al, and rewritten this part on Page 5: "Freshly exposed Fe atoms flow from underneath the DNP and wrap around it via fast surface diffusion (see Supplementary Discussions Section I). This action can be construed as Fe striving to cover up or wet carbon to lower the surface energy of the DNP. A capillary force arises from the interaction between the Fe flux and the DNP surface, directed along the DNP-
Fe interface, driving the particle towards the inside.The resultant stress at the bottom interface of Fe-DNP can reach gigapascal level, which is estimated according to the burrowing model proposed by Zimmermann et al".

Comment 3
It is well-known that small inclusions can move inside the solids driven by the gradient of chemical potential.Geguzin and Krivoglaz [2] summarize the early literature on this topic.In this respect, the atomistic Monte Carlo simulations presented in Fig. 4 just illustrate a well-known fact, and their added value is quite limited, since the employed driving forces for migration are many orders of magnitude higher than the ones relevant for the experiment.Naturally, such high driving forces result in high migration rates of several cm per sec.In fact, an exact analytical expression for the migration rate of a cubic particle can be derived, provided that the self-diffusion coefficient of Fe along the Fe-DNP interface is known [3].

Reply:
We thank the referee for the recommendation of the literature we omitted.We have cited the earlier literature at the discussion section about the thermodynamic driving force for the motion of DNPs: "We first show that there is a thermodynamic driving force available to sustain the upward mass transport of Fe.It has been reported that small inclusions can move inside the solids driven by the gradient of chemical potential 21 " (on Page 5 in the main text).
The work summarized by Geguzin and Krivoglaz about the migration of inclusions inside solids under the chemical potential gradient (Fig. 44~Fig.51 in Reference 2 "Ya.

E. Geguzin, M.A. Krivoglaz, Springer, 1973"
) are more like the well-known Kirkendall effect, in which a rigid-body mechanical motion due to the chemical potential driven lattice vacancy exchange/accumulation.Our case here has some differences.First, instead of chemically inert solid particles (or gas bubbles) as fiducial markers, here our DNP "fiducial markers" are actively dissolving, thus creating the concentration gradient in Fe crystals.Second, instead of the lattice vacancy exchange/accumulation mechanism as in the Kirkendall experiment, here it is the interfacial diffusion of Fe that generates local stresses to induce the rigid translation of particles.The Fe concentration gradient in our case is set up by carbon dissolution, driving the interfacial-diffusional flow of Fe, and subsequently the rigid translational motion of the DNP "markers".
The Monte Carlo simulation performed here is mainly to show that the DNP can move downward inside the Fe matrix, only driven by the chemical potential gradient (for example = 0.005 eV), and the motion velocity is proportional to.In experiment, the equilibrium solubility (7% at.%) of C in FCC Fe at high temperature (see Fig. S8) corresponds to the maximum chemical potential gradient of ~0.005 eV/atom across the two poles of a DNP d = 100 nm in diameter (for smaller DNPs, the value of  could be higher).It should be noted that the direct driving force for the inward motion of each DNP is the local chemical potential gradient across the nanoparticle itself instead of across the long millimeter-distance (please see the explanation in our reply to Comment 4).As for the higher chemical potential gradients μ used in the Monte Carlo simulation, we intended to show the dependence of the velocity on different chemical potential gradients.The extrapolation based on the Monte Carlo simulation suggests a v of ~25 nm/s for a DNP d = 100 nm in diameter under the maximum chemical potential gradient (~0.005 eV/atom across its two poles), comparable with the maximum experimental observation of ~50 nm/s (the DNPs reach up to ~1 mm within 5 h, the maximum motion velocity is hence ~0.2 mm/h or ~50 nm/s).Please see Supplementary discussion II and III in Supporting Information for details.In addition to the extrapolation from the direct MC simulations of DNP motion discussed above, we have also performed a back-of-the-envelope numerical estimate of the travel velocity of the DNP.The DNP is treated as a sphere with diameter d, and the Fe volume flux JcV (in the unit of m 3 /s) through the equatorial plane of the diamond sphere can be written as where dc/dz is the Fe concentration gradient (or the negative carbon concentration gradient, in units of atoms/m 3 /m) across the DNP, is the effective diffusivity of Fe atoms at the Fe-DNP interface, s is the thickness of Fe atoms flux and equals the diameter of a Fe atom, about 0.25 nm.The translational movement of the DNP fills in the space left by leaving Fe atoms, and the volume flux of iron atoms JV can also be written as where the translation velocity  can be expressed as: The above estimation indicates that a DNP of 100 nm in size could move with the velocities observed in the experiment under the action of the local chemical potential gradient across the DNP.The consistence in the maximum velocity from the experimental observation and the analytical calculations based on the driving force of the carbon concentration gradient also suggests that the large enough chemical potential difference across the DNP can be reached in experiment.

Comment 4
Following the previous comment, the gradient of carbon content in the near-surface layer is essential for the mechanism of DNPs migration proposed by the Authors.The idea seems disputable to me, since each DNP by itself is a source of carbon, which should level off any externally imposed concentration gradient.Therefore, the Authors should provide an experimental proof of varying carbon concentration in the nearsurface layer.Since the quantification of carbon content with EDS in TEM may be challenging, the atomic probe tomography (APT) seems to be better suited for this task.

Reply:
We thank the referee for the valuable suggestion.Comment 5 The Authors have to discuss alternative mechanisms of DNPs sinking into iron.For example, it is well-known that small particles can be dragged by moving grain boundaries [4], so that the recrystallization and grain growth in the iron sample may promote the spreading of DNPs into the sample interior.References:

Fig. R1 (
Fig. R1 (a) Snapshots from the in-situ TEM video showing the downward motion of DNPs into the Fe matrix; (b)The TEM image of the thinned sample after in-situ heating experiment, showing the engulfed DNP inside the Fe matrix.
has been known that diamond can be etched by Fe, Ni and Co etc. via the metalcatalyst-assisted graphitization of diamond and the carbon dissolution in metals.In our work, the Fe-catalyzed surface graphitization of DNPs is utilized for the formation of the easy diffusion channel at the Fe-diamond interface, and the gradually dissolved carbon atoms from the graphite surface layers sets up the chemical concentration gradient as the driving force for the rigid motion of DNPs.Our additional experiments (see Fig. R2) have shown that similar motion of DNPs as whole can occur inside the Ni crystals as well.The introduction of DNPs into Fe or Ni matrix here is very different from the Fe-or Ni-catalyzed diamond etching for patterning or creating nanopores in an H2 atmosphere at high temperatures.The reported diamond etching mechanism mainly involves carbon dissolution in metal Fe, Ni etc., diffusional transport to the metal-gas interface and carbon desorption in the form of methane.(Ralchenko, et al.Catalytic interaction of Fe, Ni and Pt with diamond films: patterning applications.Diamond & Related Materials 2.5-7(1993):904-909; Mehedi et al.Etching mechanism of diamond by Ni nanoparticles for fabrication of nanopores, Carbon 59(2013)448-456; Catalytic interaction of Fe, Ni and Pt with diamond films: patterning applications).

Fig. R2 .
Fig. R2.The representative Raman spectrum acquired at the ~50 m depth of the Ni-DNPs sample after the top surface ground and etched away.The inset SEM image showing the exposed DNPs after the Ni matrix was deeply etched using acid solution.

Firstly, we make
an explanation about how the local chemical concentration gradient across individual DNPs is built up and sustained.The source of C is the graphitized surface layer of each and every DNP gradually dissolving.During or even before the initial sink-in of DNPs, the surface graphitization of diamond followed by the carbon dissolution in Fe has occurred.The high enough carbon concentration in the nearsurface region could be maintained for a long time via the continual supply from these DNPs (the number of DNPs entering Fe is sufficiently large), and the concentration gradient is built up in the meantime.As each engulfed DNP with a gradually graphitized and meanwhile dissolving interfacial layer moves, the carbon concentration gradient extends forward along with them (the inhomogeneous concentration field).The driving force for the inward motion of each DNP is the local chemical potential gradient across the nanoparticle itself instead of across the long millimeter-distance (we have redrawn the schematic diagram in Fig.4ato illustrate this).Take the leading DNP as an example, as it moves forward deeper and deeper, it keeps encountering iron interior with less carbon solute content (i.e., the DNP always sees a higher Fe concentration at the location in front of its moving trajectory).The dissolved carbon atoms at the location underneath the DNP diffuse into the "virgin" iron more rapidly than into the alreadycarbon-enriched locations above the DNP.That is, in its surroundings the DNP is always accompanied by a local concentration gradient.As a result, the carbon concentration gradient extends forward along with the moving DNPs (inhomogeneous concentration field), and makes the long-distance travel of leading DNPs possible.A carbon concentration gradient in the near-surface layer has been evidenced by the depth profiles and three-dimensional-compositional images of C and Fe from the timeof-flight secondary-ion mass spectroscopy analysis (ToF-SIMS, Fig.4b).The ToF-SIMS is a surface sensitive analytical method and can demonstrate the carbon concentration gradient across a much larger distance compared with APT.The C profile in a local (much smaller volume) region, likely comes from the local concentration gradient around some DNPs, as displayed in the inset of the TOF-SIMS spectra, Fig.4b.We also have tried to characterize the local carbon profile across a single DNP quantitatively, but unfortunately, we didn't make it.We had little chance to include the leading diamond nanoparticles (their number density is quite low) inside a tiny APT needle sample (non-site-specific sample preparation via the electrochemical polishing).As for the reason why we didn't use the site-specific APT specimen preparation (FIBmillings and lift-out) to locate the individual diamond nanoparticles inside Fe matrix, because it has been well-documented that the carbon profile versus depth in Fe matrix or steels can be obviously altered by the high-energy focused ion beam irradiation (Wang, Jing, et al.Scientific Reports 7.1 (2017): 15813.;Basa et al.Metall Mater Trans A 45, 1189-1198, 2014) .

Fig. 4 
Fig. 4  Mechanism for the inward motion of DNPs into the iron crystal.(a) Schematic depicting the steps involved in the process.(b) ToF-SIMS analysis of a sample heated in furnace for 1 hour (all dispersed DNPs have entered the iron matrix) and then quenched in water to retain the carbon distribution at high temperatures as much as possible.The ToF-SIMS spectra showing the depth profiles of secondary ions of C -(grey) and Fe -(blue) in the sputtered volume from the top surface to a ~60 m depth over an area of ~5050 m 2 .The corresponding 3D images of the depth profiles visualizing the opposite concentration gradients of C and Fe.The inset ToF-SIMS spectrum displays the depth profile of C -in a much smaller volume (the analyzed area is ~55 m 2 and the depth is ~ 500 nm).